Classical and quantum bounded depth approximation algorithms

We consider some classical and quantum approximate optimization algorithms with bounded depth. First, we define a class of "local" classical optimization algorithms and show that a single step version of these algorithms can achieve the same performance as the single step QAOA on MAX-3-LIN-2. Second, we show that this class of classical algorithms generalizes a class previously considered in the literature\cite{hirvonen2014large}, and also that a single step of the classical algorithm will outperform the single-step QAOA on all triangle-free MAX-CUT instances. In fact, for all but 4 choices of degree, existing single-step classical algorithms already outperform the QAOA on these graphs, while for the remaining 4 choices we show that the generalization here outperforms it. Finally, we consider the QAOA and provide strong evidence that, for any fixed number of steps, its performance on MAX-3-LIN-2 on bounded degree graphs cannot achieve the same scaling as can be done by a class of ``global" classical algorithms. These results suggest that such local classical algorithms are likely to be at least as promising as the QAOA for approximate optimization.

Vagus nerve stimulation for chronic intractable hiccups

✓ Intractable hiccups are debilitating and usually a result of some underlying disease. Initial management includes vagal maneuvers and pharmacotherapy. When hiccups persist despite medical therapy, surgical intervention rarely is pursued. Cases described in the literature cite successful phrenic nerve blockade, crush injury, or percutaneous phrenic nerve pacing. The authors report on a case of intractable hiccups occurring after a posterior fossa stroke. Complete resolution of the spasms has been achieved to date following the placement of a vagus nerve stimulator.